The aim of this paper is to develop a preconditioned framework to deal with linear subproblems for the nonlinear and nonconvex alternating minimization algorithms, while applying to some nonconvex half quadratic regularized problems or the Ambrosio-Tortorelli approximate Mumford-Shah model [1]}, [2]}. For the half quadratic models, we mainly focus on the truncated quadratic model, the Geman-McClure model, and the Hebert-Leahy model [3]}, [4]}, [5]}, [6]}. The truncated quadratic models including the Geman-Reynolds type model [3]} and Geman-Yang type model [4]} are rooted from the Markov random fields.
All the half quadratic models addressed here have statistical interpretations and we refer to [9]}, [10]}, [11]} for more details.
These nonconvex regularizations are widely used for image restorations, segmentation, stereo, optical flow, and so on, which have vast applications in medical imaging, computer vision, and inverse problems [12]}, [9]}, [14]}, [11]}, [16]}, [2]}.
